pith. sign in

arxiv: 2506.11552 · v2 · submitted 2025-06-13 · 🪐 quant-ph · cs.LG

Learning Encodings by Maximizing State Distinguishability: Variational Quantum Error Correction

Nico Meyer , Christopher Mutschler , Andreas Maier , Daniel D. Scherer This is my paper

Pith reviewed 2026-05-19 09:54 UTC · model grok-4.3

classification 🪐 quant-ph cs.LG
keywords quantum error correctionvariational quantum algorithmsmachine learning for quantumquantum noise modelsstate distinguishabilityencoding optimizationnear-term quantum devices
0
0 comments X

The pith

By maximizing distinguishability of states after noise, variational circuits learn efficient quantum error correction encodings tailored to specific noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a new objective for designing quantum error correction codes that maximizes how distinguishable encoded states remain after experiencing noise. This distinguishability loss is used as a machine learning objective in variational quantum circuits to find encoding circuits optimized for given noise characteristics. The resulting VarQEC method produces codes with lower overhead than traditional ones like the surface code. It is shown to outperform standard codes in various scenarios and has been tested on actual quantum hardware from IBM and IQM.

Core claim

The central claim is that optimizing encodings by maximizing state distinguishability after the noise channel leads to resource-efficient codes that support efficient recovery operations and outperform conventional quantum error correction codes for specific noise structures.

What carries the argument

The distinguishability loss function, which measures the ability to distinguish between different logical states following the application of a noise channel, serving as the variational objective for training encoding circuits.

Load-bearing premise

That maximizing the distinguishability between states after the noise channel is enough to ensure that efficient recovery operations can achieve high fidelity.

What would settle it

A direct comparison experiment where the learned codes fail to achieve higher fidelity or require more resources than standard codes like the surface code under the same noise conditions.

Figures

Figures reproduced from arXiv: 2506.11552 by Andreas Maier, Christopher Mutschler, Daniel D. Scherer, Nico Meyer.

Figure 1
Figure 1. Figure 1: We propose a procedure that tailors quantum [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Effect of noise on a physical state (a) and [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pipeline of learning VarQEC encodings with the distinguishability loss. Pairs of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An instance of the randomized entangling [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Distinguishability loss for different QEC codes on (a) depolarizing and (b) asymmetric depolarizing noise. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Stability and resilience of standard QEC and [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Fidelity loss for different QEC codes and respective recovery operations on (a) depolarizing and (b) [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Experiment on the ibm_marrakesh de￾vice [79], which features a 156-qubit Heron r2 chip. We induce thermal relaxation noise by applying a delay on all wires for a specific duration, assuming a median T1 = 180µs and T2 = 120µs. The VarQEC codes are trained in classical simulation assuming these noise hy￾perparameters. Evaluation is conducted on hardware si￾multaneously on 8 logical patches distributed unifor… view at source ↗
Figure 9
Figure 9. Figure 9: Empirical analysis of how many Haar-random states are necessary to get a stable approximation of the [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Empirical analysis on how well the two-design formulations approximate the groundtruth of the distin [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Empirical analysis on how a weighted two-design can reduce the approximation bias, compared to a [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The adapted QVECTOR pipeline [46] for training recovery operations Urec(Φ), under the premise that the encoding operations Uenc(Θ) have already been trained with the distinguishability loss. The average-case fidelity loss FS is evaluated on states from a two-design S and is minimized w.r.t. the free parameters of the recovery operation. We note that at the end of the circuits it is not guaranteed that the… view at source ↗
Figure 13
Figure 13. Figure 13: An instance of the randomized entangling ansatz (RAG) on [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Distinguishability loss for different QEC codes on (a) bit-flip, (b) thermal relaxation, (c) amplitude [PITH_FULL_IMAGE:figures/full_fig_p040_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Fidelity loss for different QEC codes and respective recovery operations on (a) bit-flip, (b) thermal [PITH_FULL_IMAGE:figures/full_fig_p041_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Distinguishability loss for different QEC codes with two logical qubits, i.e. [PITH_FULL_IMAGE:figures/full_fig_p043_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Stability of standard QEC and VarQEC codes with increasing noise strength for (a) symmetric and (b) [PITH_FULL_IMAGE:figures/full_fig_p044_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Stability of standard QEC and VarQEC codes in setups of varying (a) noise strength on different qubits [PITH_FULL_IMAGE:figures/full_fig_p045_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Different layouts for n = 5 physical qubits satisfying the connectivity restrictions of some potential underlying hardware. Throughout most of this paper, we assumed a “full” connectivity, which is not available on most hardware systems. Similarly, the “dense” layout might be hard to realize with on most platforms. The “square” as well as the “star” layout aligns e.g. with the grid-based topology of the G… view at source ↗
Figure 20
Figure 20. Figure 20: Resilience of standard QEC and VarQEC codes in setups of non-ideal encoding operations under symmetric [PITH_FULL_IMAGE:figures/full_fig_p046_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Distinguishability loss for different QEC codes on thermal relaxation noise modeling the median error [PITH_FULL_IMAGE:figures/full_fig_p047_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Reconstructed distinguishability loss for the [PITH_FULL_IMAGE:figures/full_fig_p048_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Circuit ansatz implementing the [[4, 1]] VarQEC code. The parameters were trained on the noise setup for the iqm_marrakesh experiment that is analyzed in [PITH_FULL_IMAGE:figures/full_fig_p049_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Reconstructed distinguishability loss for the [PITH_FULL_IMAGE:figures/full_fig_p049_24.png] view at source ↗
read the original abstract

Quantum error correction is crucial for protecting quantum information against decoherence. Traditional codes like the surface code require substantial overhead, making them impractical for near-term, early fault-tolerant devices. We propose a novel objective function for tailoring error correction codes to specific noise structures by maximizing the distinguishability between quantum states after a noise channel, ensuring efficient recovery operations. We formalize this concept with the distinguishability loss function, serving as a machine learning objective to discover resource-efficient encoding circuits optimized for given noise characteristics. We implement this methodology using variational techniques, termed variational quantum error correction (VarQEC). Our approach yields codes with desirable theoretical and practical properties and outperforms standard codes in various scenarios. We also provide proof-of-concept demonstrations on IBM and IQM hardware devices, highlighting the practical relevance of our procedure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces variational quantum error correction (VarQEC), a method that learns encoding circuits by variationally maximizing a distinguishability loss function computed after a given noise channel. The goal is to produce resource-efficient codes tailored to specific noise models, with claims that the resulting encodings support efficient recovery, exhibit desirable theoretical and practical properties, and outperform standard codes such as the surface code in various scenarios. Proof-of-concept demonstrations on IBM and IQM hardware are provided.

Significance. If the distinguishability objective can be shown to control logical fidelity under an explicit recovery map, the approach would offer a practical, noise-adaptive alternative to fixed codes for near-term devices. The variational formulation and hardware demonstrations are strengths that could enable tailored error correction with lower overhead than conventional constructions.

major comments (2)
  1. [§3] §3 (Method): The distinguishability loss is defined directly from the noise channel, yet no derivation or inequality is supplied that bounds the diamond-norm distance of the effective logical channel to the identity or guarantees that a simple (e.g., syndrome-based) recovery map achieves high fidelity. Without this link the central claim that maximization yields efficient, high-fidelity recovery remains unestablished.
  2. [§4] §4 (Numerical experiments): The reported outperformance over standard codes is stated without tabulated logical error rates, fidelity values, or statistical error bars for the chosen noise models and code distances; the absence of these quantitative metrics prevents verification of the performance advantage.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the noise model, number of shots, and variational circuit depth used in each hardware run.
  2. [Abstract] The abstract asserts that the method 'ensures efficient recovery operations' without qualifying that this is an empirical observation rather than a proven guarantee.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our work. We address each of the major comments below and outline the changes we will make to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Method): The distinguishability loss is defined directly from the noise channel, yet no derivation or inequality is supplied that bounds the diamond-norm distance of the effective logical channel to the identity or guarantees that a simple (e.g., syndrome-based) recovery map achieves high fidelity. Without this link the central claim that maximization yields efficient, high-fidelity recovery remains unestablished.

    Authors: We acknowledge the validity of this comment. The current manuscript motivates the distinguishability loss as a proxy for enabling effective recovery by maximizing the separation of logical states after the noise channel, but does not provide a formal inequality linking it directly to the diamond norm of the logical channel or to the performance of a specific recovery map. This connection is indeed important for rigorously establishing the central claim. In the revised manuscript, we will expand §3 to include a theoretical discussion that relates the distinguishability loss to bounds on the logical error, for instance by connecting it to the trace distance between the noisy encoded states and discussing implications for syndrome-based recovery. We will also note the heuristic nature of the approach while emphasizing the supporting numerical results. revision: yes

  2. Referee: [§4] §4 (Numerical experiments): The reported outperformance over standard codes is stated without tabulated logical error rates, fidelity values, or statistical error bars for the chosen noise models and code distances; the absence of these quantitative metrics prevents verification of the performance advantage.

    Authors: We agree that providing detailed quantitative metrics would enhance the verifiability of our results. The manuscript presents performance comparisons primarily through figures, but we recognize the need for tabulated data. In the revised version, we will include a new table in §4 that reports the logical error rates and average fidelities for the VarQEC encodings versus standard codes like the surface code, across the considered noise models and distances. We will also add statistical error bars based on multiple independent optimization runs to quantify the variability. revision: yes

Circularity Check

0 steps flagged

Distinguishability loss defined from external noise channel; variational optimization yields encodings without reduction to fitted inputs by construction.

full rationale

The derivation begins with an externally specified noise channel and defines a distinguishability loss directly from post-channel state overlaps. This loss is then maximized variationally to obtain an encoding circuit. Numerical demonstrations on specific noise models and hardware are presented as evidence of performance, rather than any claim that the loss mathematically forces recovery fidelity by definition. No equations reduce a claimed prediction to a parameter fitted inside the same paper, and no load-bearing uniqueness theorem or ansatz is imported solely via self-citation. The central construction therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields limited visibility into internal assumptions. The central claim rests on the unproven premise that distinguishability maximization produces recoverable codes and on the existence of a variational landscape that can be optimized to useful minima.

free parameters (1)
  • variational circuit parameters
    Circuit angles or gate parameters are adjusted by the optimizer to minimize the distinguishability loss; their final values are fitted to the chosen noise model.
axioms (1)
  • domain assumption Maximizing post-noise distinguishability implies the existence of an efficient recovery map
    Invoked when the abstract states that the loss 'ensures efficient recovery operations' without further derivation.

pith-pipeline@v0.9.0 · 5669 in / 1285 out tokens · 28852 ms · 2026-05-19T09:54:00.262735+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Learning to Concatenate Quantum Codes

    quant-ph 2026-04 unverdicted novelty 6.0

    A machine-learning approach adaptively chooses quantum code sequences for concatenation to achieve target logical error rates with far fewer qubits than standard methods for structured noise.

  2. A Review of Variational Quantum Algorithms: Insights into Fault-Tolerant Quantum Computing

    quant-ph 2026-04 unverdicted novelty 1.0

    A literature review of VQAs covering ansatz design, classical optimization, barren plateaus, error mitigation strategies, and theoretical adaptations for fault-tolerant quantum computing.

Reference graph

Works this paper leans on

113 extracted references · 113 canonical work pages · cited by 2 Pith papers · 7 internal anchors

  1. [1]

    Challenges and opportunities in quantum optimiza- tion

    Amira Abbas, Andris Ambainis, Bran- don Augustino, Andreas Bärtschi, Harry Buhrman, Carleton Coffrin, Giorgio Cor- tiana, Vedran Dunjko, Daniel J Egger, Bruce G Elmegreen, et al. Challenges and opportunities in quantum optimiza- tion. Nat. Rev. Phys., pages 1–18, 2024. DOI: 10.1038/s42254-024-00770-9

    work page doi:10.1038/s42254-024-00770-9 2024

  2. [2]

    Technology and per- formance benchmarks of iqm’s 20-qubit quantum com- puter,

    Leonid Abdurakhimov, Janos Adam, Has- nain Ahmad, Olli Ahonen, Manuel Al- gaba, Guillermo Alonso, Ville Bergholm, Rohit Beriwal, Matthias Beuerle, Clinton Bockstiegel, et al. Technology and perfor- mance benchmarks of iqm’s 20-qubit quan- tum computer. arXiv:2408.12433, 2024. DOI: 10.48550/arXiv.2408.12433

    work page doi:10.48550/arxiv.2408.12433 2024

  3. [3]

    Gupta, Neereja Sundaresan, Thomas Alexander, Christopher J

    Rajeev Acharya, Laleh Aghababaie-Beni, Igor Aleiner, Trond I Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abra- ham Asfaw, Nikita Astrakhantsev, Juan Atalaya, et al. Quantum error correction below the surface code threshold. Nature, 16 638:920–926, 2025. DOI: 10.1038/s41586- 024-08449-y

    work page doi:10.1038/s41586- 2025

  4. [4]

    Artificial intelligence for quantum computing

    Yuri Alexeev, Marwa H Farag, Taylor L Patti, Mark E Wolf, Natalia Ares, Alán Aspuru-Guzik, Simon C Benjamin, Zhenyu Cai, Zohim Chandani, Federico Fedele, et al. Artificial intelligence for quantum computing. arXiv:2411.09131, 2024. DOI: 10.48550/arXiv.2411.09131

    work page doi:10.48550/arxiv.2411.09131 2024

  5. [5]

    Fault- tolerant quantum computation against bi- ased noise

    Panos Aliferis and John Preskill. Fault- tolerant quantum computation against bi- ased noise. Phys. Rev. A, 78(5):052331,

    work page

  6. [6]

    DOI: 10.1103/PhysRevA.78.052331

    work page doi:10.1103/physreva.78.052331

  7. [7]

    Quantum t-designs: t-wise independence in the quantum world

    Andris Ambainis and Joseph Emerson. Quantum t-designs: t-wise independence in the quantum world. In Twenty-Second Annual IEEE Conference on Computa- tional Complexity (CCC’07), pages 129– 140, 2007. DOI: 10.1109/CCC.2007.26

    work page doi:10.1109/ccc.2007.26 2007

  8. [8]

    Stability of stochastic gradient descent on nonsmooth convex losses

    Raef Bassily, Vitaly Feldman, Cristóbal Guzmán, and Kunal Talwar. Stability of stochastic gradient descent on nonsmooth convex losses. In Advances in Neural In- formation Processing Systems (NeurIPS), volume 33, pages 4381–4391, 2020. DOI: 10.48550/arXiv.2006.06914

    work page doi:10.48550/arxiv.2006.06914 2020

  9. [9]

    Quan- tum codes from neural networks

    JohannesBauschandFelixLeditzky. Quan- tum codes from neural networks. New J. Phys. , 22(2):023005, 2020. DOI: 10.1088/1367-2630/ab6cdd

    work page doi:10.1088/1367-2630/ab6cdd 2020

  10. [10]

    Learning high-accuracy error decoding for quantum processors.Nature, 635:834–840, 2024

    Johannes Bausch, Andrew W Senior, Fran- cisco JH Heras, Thomas Edlich, Alex Davies, Michael Newman, Cody Jones, Kevin Satzinger, Murphy Yuezhen Niu, Sam Blackwell, et al. Learning high- accuracy error decoding for quantum pro- cessors. Nature, 635:834–840, 2024. DOI: 10.1038/s41586-024-08148-8

    work page doi:10.1038/s41586-024-08148-8 2024

  11. [12]

    PennyLane: Automatic differentiation of hybrid quantum-classical computations

    Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M Sohaib Alam, Guillermo Alonso-Linaje, B AkashNarayanan, Ali Asadi, et al. Pennylane: Automatic differ- entiation of hybrid quantum-classical com- putations. arXiv:1811.04968, 2018. DOI: 10.48550/arXiv.1811.04968

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1811.04968 2018

  12. [13]

    [Online]

    Jacob Biamonte, Peter Wittek, Nicola Pan- cotti, Patrick Rebentrost, Nathan Wiebe, and Seth Lloyd. Quantum machine learn- ing. Nature, 549(7671):195–202, 2017. DOI: 10.1038/nature23474

    work page doi:10.1038/nature23474 2017

  13. [14]

    An extension of kakutani’s theoremoninfiniteproductmeasurestothe tensor product of semifinite w*-algebras

    Donald Bures. An extension of kakutani’s theoremoninfiniteproductmeasurestothe tensor product of semifinite w*-algebras. Trans. Am. Math. Soc., 135:199–212, 1969. DOI: 10.2307/1995012

    work page doi:10.2307/1995012 1969

  14. [15]

    Good quantum error-correcting codes ex- ist

    A Robert Calderbank and Peter W Shor. Good quantum error-correcting codes ex- ist. Phys. Rev. A, 54(2):1098, 1996. DOI: 10.1103/PhysRevA.54.1098

    work page doi:10.1103/physreva.54.1098 1996

  15. [16]

    Quan- tumerrorcorrectionandorthogonalgeome- try

    A Robert Calderbank, Eric M Rains, Pe- ter W Shor, and Neil JA Sloane. Quan- tumerrorcorrectionandorthogonalgeome- try. Phys. Rev. Lett., 78(3):405, 1997. DOI: 10.1103/PhysRevLett.78.405

    work page doi:10.1103/physrevlett.78.405 1997

  16. [17]

    Ruishi Chen, Victor R Lee, Annie Camey Kuo, Denise Clark Pope, and Sarah Miles

    A Colin Cameron, Jonah B Gelbach, and Douglas L Miller. Bootstrap-based im- provements for inference with clustered er- rors. Rev. Econ. Stat., 90(3):414–427, 2008. DOI: 10.1162/rest.90.3.414

    work page doi:10.1162/rest.90.3.414 2008

  17. [18]

    Roads towards fault- tolerant universal quantum computation

    Earl T Campbell, Barbara M Terhal, and Christophe Vuillot. Roads towards fault- tolerant universal quantum computation. Nature, 549(7671):172–179, 2017. DOI: 10.1038/nature23460

    work page doi:10.1038/nature23460 2017

  18. [19]

    Quan- tum variational learning for quantum error- correcting codes

    Chenfeng Cao, Chao Zhang, Zipeng Wu, Markus Grassl, and Bei Zeng. Quan- tum variational learning for quantum error- correcting codes. Quantum, 6:828, 2022. DOI: 10.22331/q-2022-10-06-828

    work page doi:10.22331/q-2022-10-06-828 2022

  19. [20]

    Variational Quantum Algorithms,

    Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quantum algorithms.Na- ture Rev. Phys., 3(9):625–644, 2021. DOI: 10.1038/s42254-021-00348-9

    work page doi:10.1038/s42254-021-00348-9 2021

  20. [21]

    Cost function dependent barren plateaus in shallow parametrized quantum circuits,

    Marco Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J Coles. Cost function dependent barren plateaus in shal- low parametrized quantum circuits. Na- ture Commun., 12(1):1791, 2021. DOI: 10.1038/s41467-021-21728-w

    work page doi:10.1038/s41467-021-21728-w 2021

  21. [22]

    Cerezo, M

    Marco Cerezo, Martin Larocca, Diego García-Martín, Nelson L Diaz, Paolo Brac- 17 cia, Enrico Fontana, Manuel S Rudolph, Pablo Bermejo, Aroosa Ijaz, Supanut Thanasilp, et al. Does provable absence of barren plateaus imply classical simula- bility? arXiv:2312.09121, 2023. DOI: 10.48550/arXiv.2312.09121

    work page doi:10.48550/arxiv.2312.09121 2023

  22. [23]

    Flag fault-tolerant error correc- tion with arbitrary distance codes.Quan- tum, 2:53, 2018

    Christopher Chamberland and Michael E Beverland. Flag fault-tolerant error correc- tion with arbitrary distance codes.Quan- tum, 2:53, 2018. DOI: 10.22331/q-2018-02- 08-53

    work page doi:10.22331/q-2018-02- 2018

  23. [24]

    Automated discovery of logical gates for quantum error correction

    Hongxiang Chen, Michael Vasmer, Niko- las P Breuckmann, and Edward Grant. Automated discovery of logical gates for quantum error correction. Quantum Inf. Comput., 22(11-12):947–964, 2022. DOI: 10.26421/QIC22.11-12-3

    work page doi:10.26421/qic22.11-12-3 2022

  24. [25]

    Quan- tum autoencoders for efficient compression of quantum data,

    Ranyiliu Chen, Zhixin Song, Xuanqiang Zhao, and Xin Wang. Variational quan- tum algorithms for trace distance and fi- delity estimation. Quantum Sci. Technol., 7(1):015019, 2021. DOI: 10.1088/2058- 9565/ac38ba

    work page doi:10.1088/2058- 2021

  25. [26]

    Quantum convolutional neural net- works

    Iris Cong, Soonwon Choi, and Mikhail D Lukin. Quantum convolutional neural net- works. Nat. Phys., 15(12):1273–1278, 2019. DOI: 10.1038/s41567-019-0648-8

    work page doi:10.1038/s41567-019-0648-8 2019

  26. [27]

    In- formation theory: coding theorems for discrete memoryless systems

    Imre Csiszár and János Körner. In- formation theory: coding theorems for discrete memoryless systems . Cam- bridge University Press, 2011. DOI: 10.1017/CBO9780511921889

    work page doi:10.1017/cbo9780511921889 2011

  27. [28]

    Practical quantum advantage in quantum simulation

    Andrew J Daley, Immanuel Bloch, Chris- tian Kokail, Stuart Flannigan, Natalie Pearson, Matthias Troyer, and Peter Zoller. Practical quantum advantage in quantum simulation. Nature, 607(7920):667–676,

    work page

  28. [29]

    DOI: 10.1038/s41586-022-04940-6

    work page doi:10.1038/s41586-022-04940-6

  29. [31]

    The computational power of random quantum circuits in arbitrary geometries (2024)

    Matthew DeCross, Reza Haghshenas, Minzhao Liu, Enrico Rinaldi, Johnnie Gray, Yuri Alexeev, Charles H Baldwin, John P Bartolotta, Matthew Bohn, Eli Chertkov, et al. The computational power of random quantum circuits in arbitrary geometries. arXiv:2406.02501, 2024. DOI: 10.48550/arXiv.2406.02501

    work page doi:10.48550/arxiv.2406.02501 2024

  30. [32]

    Deshpande, M

    Abhinav Deshpande, Marcel Hinsche, Sona Najafi, Kunal Sharma, Ryan Sweke, and Christa Zoufal. Dynamic parameterized quantum circuits: expressive and barren- plateau free.arXiv:2411.05760, 2024. DOI: 10.48550/arXiv.2411.05760

    work page doi:10.48550/arxiv.2411.05760 2024

  31. [35]

    Cryptographic distinguishability measures for quantum-mechanical states

    Christopher A Fuchs and Jeroen Van De Graaf. Cryptographic distinguishability measures for quantum-mechanical states. IEEE Trans. Inf. Theory, 45(4):1216–1227,

    work page

  32. [36]

    DOI: 10.1109/18.761271

    work page doi:10.1109/18.761271

  33. [37]

    Quantum simulation

    Iulia M Georgescu, Sahel Ashhab, and Franco Nori. Quantum simulation. Rev. Mod. Phys., 86(1):153–185, 2014. DOI: 10.1103/RevModPhys.86.153

    work page internal anchor Pith review doi:10.1103/revmodphys.86.153 2014

  34. [38]

    Stabilizer codes and quantum error correction

    Daniel Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, 1997. URL https://thesis.library.caltech. edu/2900/2/THESIS.pdf

    work page 1997

  35. [39]

    An introduction to quantum error correction

    Daniel Gottesman. An introduction to quantum error correction. In Proc. Sym. Appl. Math., volume 58, pages 221–236,

    work page

  36. [40]

    DOI: 10.1090/psapm/068

    work page doi:10.1090/psapm/068

  37. [41]

    Nature Communications , volume =

    Harper R Grimsley, Sophia E Economou, Edwin Barnes, and Nicholas J Mayhall. An adaptive variational algorithm for exact molecular simulations on a quantum com- puter. Nature Commun., 10(1):3007, 2019. DOI: 10.1038/s41467-019-10988-2

    work page doi:10.1038/s41467-019-10988-2 2019

  38. [42]

    Properties of the Quantum Channel

    Laszlo Gyongyosi and Sandor Imre. Properties of the quantum chan- nel. arXiv:1208.1270, 2012. DOI: 10.48550/arXiv.1208.1270

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1208.1270 2012

  39. [43]

    In: Proc

    Trevor Hastie. The elements of statistical learning: data mining, inference, and pre- 18 diction. Springer, 2009. DOI: 10.1007/978- 0-387-84858-7

    work page doi:10.1007/978- 2009

  40. [44]

    Discov- ering highly efficient low-weight quantum error-correcting codes with reinforcement learning

    Austin Yubo He and Zi-Wen Liu. Discov- ering highly efficient low-weight quantum error-correcting codes with reinforcement learning. arXiv:2502.14372, 2025. DOI: 10.48550/arXiv.2502.14372

    work page doi:10.48550/arxiv.2502.14372 2025

  41. [46]

    The proper formula for relati ve entropy and its asymptotics in quantum probability

    Fumio Hiai and Dénes Petz. The proper formula for relative entropy and its asymp- totics in quantum probability. Comm. Math. Phys., 143:99–114, 1991. DOI: 10.1007/BF02100287

    work page doi:10.1007/bf02100287 1991

  42. [47]

    Sur- face code quantum computing by lattice surgery

    Dominic Horsman, Austin G Fowler, Si- mon Devitt, and Rodney Van Meter. Sur- face code quantum computing by lattice surgery.New J. Phys., 14(12):123011, 2012. DOI: 10.1088/1367-2630/14/12/123011

    work page doi:10.1088/1367-2630/14/12/123011 2012

  43. [48]

    Discovery of an exchange-only gate sequence for cnot with record-low gate time using reinforcement learning

    Violeta N Ivanova-Rohling, Niklas Rohling, and Guido Burkard. Discovery of an exchange-only gate sequence for cnot with record-low gate time using reinforcement learning. arXiv:2402.10559, 2024. DOI: 10.48550/arXiv.2402.10559

    work page doi:10.48550/arxiv.2402.10559 2024

  44. [49]

    Quantum computing with Qiskit

    Ali Javadi-Abhari, Matthew Treinish, Kevin Krsulich, Christopher J Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D Nation, Lev S Bishop, Andrew W Cross, et al. Quantum computing with Qiskit. arXiv:2405.08810, 2024. DOI: 10.48550/arXiv.2405.08810

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2405.08810 2024

  45. [50]

    QVECTOR: an algorithm for device-tailored quantum error correction

    Peter D Johnson, Jonathan Romero, Jonathan Olson, Yudong Cao, and Alán Aspuru-Guzik. Qvector: an algorithm for device-tailored quantum error correc- tion. arXiv:1711.02249, 2017. DOI: 10.48550/arXiv.1711.02249

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1711.02249 2017

  46. [51]

    Fidelity for mixed quantum states

    Richard Jozsa. Fidelity for mixed quantum states. J. Mod. Opt., 41(12):2315–2323,

    work page

  47. [52]

    DOI: 10.1080/09500349414552171

    work page doi:10.1080/09500349414552171

  48. [53]

    Qiskit experiments: A python pack- age to characterize and calibrate quantum computers

    NaokiKanazawa, DanielJEgger, YaelBen- Haim, Helena Zhang, William E Shanks, Gadi Aleksandrowicz, and Christopher J Wood. Qiskit experiments: A python pack- age to characterize and calibrate quantum computers. J. Open Source Softw., 8(84): 5329, 2023. DOI: 10.21105/joss.05329

    work page doi:10.21105/joss.05329 2023

  49. [54]

    Katabarwa, K

    Amara Katabarwa, Katerina Gratsea, Athena Caesura, and Peter D Johnson. Early fault-tolerant quantum computing. PRX Quantum, 5(2):020101, 2024. DOI: 10.1103/PRXQuantum.5.020101

    work page doi:10.1103/prxquantum.5.020101 2024

  50. [55]

    Concatenated Quantum Codes

    Emanuel Knill and Raymond Laflamme. Concatenated quantum codes. arXiv:quant-ph/9608012, 1996. DOI: 10.48550/arXiv.quant-ph/9608012

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.quant-ph/9608012 1996

  51. [56]

    Theory of quantum error-correcting codes

    Emanuel Knill and Raymond Laflamme. Theory of quantum error-correcting codes. Phys. Rev. A , 55(2):900, 1997. DOI: 10.1103/PhysRevA.55.900

    work page doi:10.1103/physreva.55.900 1997

  52. [57]

    General state changes in quan- tum theory

    Karl Kraus. General state changes in quan- tum theory. Ann. Phys., 64(2):311–335,

    work page

  53. [58]

    DOI: 10.1016/0003-4916(71)90108-4

    work page doi:10.1016/0003-4916(71)90108-4

  54. [59]

    Perfect quantum error correcting code

    Raymond Laflamme, Cesar Miquel, Juan Pablo Paz, and Wojciech Hubert Zurek. Perfect quantum error correcting code. Phys. Rev. Lett., 77(1):198, 1996. DOI: 10.1103/PhysRevLett.77.198

    work page doi:10.1103/physrevlett.77.198 1996

  55. [60]

    Autonomous quantum er- ror correction and quantum computa- tion

    José Lebreuilly, Kyungjoo Noh, Chiao- Hsuan Wang, Steven M Girvin, and Liang Jiang. Autonomous quantum er- ror correction and quantum computa- tion. arXiv:2103.05007, 2021. DOI: 10.48550/arXiv.2103.05007

    work page doi:10.48550/arxiv.2103.05007 2021

  56. [62]

    Foundations and trends in multimodal machine learning: Princi- ples, challenges, and open questions.ACM Comput

    Paul Pu Liang, Amir Zadeh, and Louis- Philippe Morency. Foundations and trends in multimodal machine learning: Princi- ples, challenges, and open questions.ACM Comput. Surv., 56(10):1–42, 2024

    work page 2024

  57. [63]

    Completely positive maps and entropy inequalities

    Göran Lindblad. Completely positive maps and entropy inequalities. Comm. Math. Phys., 40:147–151, 1975. DOI: 10.1007/BF01609396

    work page doi:10.1007/bf01609396 1975

  58. [64]

    On the limited memory BFGS method for large scale optimization

    Dong C Liu and Jorge Nocedal. On the limited memory bfgs method for large scale optimization. Math. Program., 45(1):503– 528, 1989. DOI: 10.1007/BF01589116

    work page doi:10.1007/bf01589116 1989

  59. [65]

    Noise-strength-adapted approx- imate quantum codes inspired by machine 19 learning

    Shuwei Liu, Shiyu Zhou, Zi-Wen Liu, and Jinmin Yi. Noise-strength-adapted approx- imate quantum codes inspired by machine 19 learning. arXiv:2503.11783, 2025. DOI: 10.48550/arXiv.2503.11783

    work page doi:10.48550/arxiv.2503.11783 2025

  60. [66]

    Quantum error correction with quantum autoencoders

    David F Locher, Lorenzo Cardarelli, and Markus Müller. Quantum error correction with quantum autoencoders. Quantum, 7: 942, 2023. DOI: 10.22331/q-2023-03-09- 942

    work page doi:10.22331/q-2023-03-09- 2023

  61. [67]

    Optimization of tensor network codes with reinforcement learning

    Caroline Mauron, Terry Farrelly, and Thomas M Stace. Optimization of tensor network codes with reinforcement learning. New J. Phys., 26(2):023024, 2024. DOI: 10.1088/1367-2630/ad23a6

    work page doi:10.1088/1367-2630/ad23a6 2024

  62. [68]

    2018 , month =

    Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hart- mut Neven. Barren plateaus in quan- tum neural network training landscapes. Nat. Commun., 9(1):4812, 2018. DOI: 10.1038/s41467-018-07090-4

    work page doi:10.1038/s41467-018-07090-4 2018

  63. [69]

    Bench- marking quantum processor performance at scale,

    David C McKay, Ian Hincks, Emily J Pritchett, Malcolm Carroll, Luke CG Govia, and Seth T Merkel. Benchmark- ing quantum processor performance at scale. arXiv:2311.05933, 2023. DOI: 10.48550/arXiv.2311.05933

    work page doi:10.48550/arxiv.2311.05933 2023

  64. [70]

    Meyer, C

    Nico Meyer, Christian Ufrecht, Manira- man Periyasamy, Daniel D Scherer, Axel Plinge, and Christopher Mutschler. A survey on quantum reinforcement learn- ing. arXiv:2211.03464, 2022. DOI: 10.48550/arXiv.2211.03464

    work page doi:10.48550/arxiv.2211.03464 2022

  65. [71]

    Dubey, Christian Ufrecht, Maniraman Periyasamy, Axel Plinge, Christopher Mutschler & Daniel D

    Nico Meyer, Jakob Murauer, Alexan- der Popov, Christian Ufrecht, Axel Plinge, Christopher Mutschler, and Daniel D Scherer. Warm-start varia- tional quantum policy iteration. In IEEE International Conference on Quantum Computing and Engineering (QCE), vol- ume 1, pages 1458–1466, 2024. DOI: 10.1109/QCE60285.2024.00172

    work page doi:10.1109/qce60285.2024.00172 2024

  66. [72]

    Dubey, Christian Ufrecht, Maniraman Periyasamy, Axel Plinge, Christopher Mutschler & Daniel D

    Nico Meyer, Christian Ufrecht, Manira- man Periyasamy, Axel Plinge, Christopher Mutschler, Daniel D Scherer, and Andreas Maier. Qiskit-torch-module: Fast proto- typing of quantum neural networks. In IEEE International Conference on Quan- tum Computing and Engineering (QCE), volume 1, pages 817–823, 2024. DOI: 10.1109/QCE60285.2024.00101

    work page doi:10.1109/qce60285.2024.00101 2024

  67. [73]

    Bench- marking quantum reinforcement learn- ing

    Nico Meyer, Christian Ufrecht, George Yammine, Georgios Kontes, Christopher Mutschler, and Daniel D Scherer. Bench- marking quantum reinforcement learn- ing. arXiv:2501.15893, 2025. DOI: 10.48550/arXiv.2501.15893

    work page doi:10.48550/arxiv.2501.15893 2025

  68. [74]

    How to gener- ate random matrices from the clas- sical compact groups

    Francesco Mezzadri. How to gener- ate random matrices from the clas- sical compact groups. Not. Am. Math. Soc., 54(5):592–604, 2006. URL https://www.ams.org/notices/200705/ fea-mezzadri-web.pdf

    work page 2006

  69. [75]

    Optimizing quantum error correction codes with reinforcement learning

    HendrikPoulsenNautrup, NicolasDelfosse, Vedran Dunjko, Hans J Briegel, and Nicolai Friis. Optimizing quantum error correction codes with reinforcement learning. Quan- tum, 3:215, 2019. DOI: 10.22331/q-2019- 12-16-215

    work page doi:10.22331/q-2019- 2019

  70. [76]

    Quantum Computation and Quantum In- formation

    Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum In- formation. Cambridge University Press,

    work page

  71. [77]

    DOI: 10.1017/CBO9780511976667

    work page doi:10.1017/cbo9780511976667

  72. [78]

    M.; Diamanti, E.; Kerenidis, I

    Jan Olle, Remmy Zen, Matteo Puviani, and Florian Marquardt. Simultaneous dis- covery of quantum error correction codes and encoders with a noise-aware reinforce- ment learning agent. npj Quantum Inf., 10(126):1–17, 2024. DOI: 10.1038/s41534- 024-00920-y

    work page doi:10.1038/s41534- 2024

  73. [79]

    Scaling the automated discovery of quantum cir- cuits via reinforcement learning with gad- gets

    Jan Ollé Aguilera, Oleg M Yevtushenko, and Florian Marquardt. Scaling the automated discovery of quantum cir- cuits via reinforcement learning with gad- gets. arXiv:2503.11638, 2025. DOI: 10.48550/arXiv.2503.1163

    work page doi:10.48550/arxiv.2503.1163 2025

  74. [80]

    Hamiltonian variational ansatz without barren plateaus

    Chae-Yeun Park and Nathan Killoran. Hamiltonian variational ansatz without barren plateaus. Quantum, 8:1239, 2024. DOI: 10.22331/q-2024-02-01-1239

    work page doi:10.22331/q-2024-02-01-1239 2024

  75. [82]

    Pytorch: An impera- tive style, high-performance deep learning library

    AdamPaszke, SamGross, FranciscoMassa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Al- ban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and 20 Soumith Chintala. Pytorch: An impera- tive style, high-p...

    work page

  76. [83]

    PyTorch: An Imperative Style, High-Performance Deep Learning Library

    Curran Associates, Inc., 2019. DOI: 10.48550/arXiv.1912.01703

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1912.01703 2019

  77. [84]

    Sufficient subalgebras and the relative entropy of states of a von neumann algebra

    Dénes Petz. Sufficient subalgebras and the relative entropy of states of a von neumann algebra. Commun. Math. Phys., 105:123– 131, 1986. DOI: 10.1007/BF01212345

    work page doi:10.1007/bf01212345 1986

  78. [85]

    New mate- rial platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds

    Alexander PM Place, Lila VH Rodgers, Pranav Mundada, Basil M Smitham, Mat- tias Fitzpatrick, Zhaoqi Leng, Anjali Premkumar, Jacob Bryon, Andrei Vraji- toarea, Sara Sussman, et al. New mate- rial platform for superconducting transmon qubits with coherence times exceeding 0.3 milliseconds. Nature Commun., 12(1):1779,

    work page

  79. [86]

    DOI: 10.1038/s41467-021-22030-5

    work page doi:10.1038/s41467-021-22030-5

  80. [87]

    Qdistrnd: A gap package for computing the distance of quantum error-correcting codes

    Leonid P Pryadko, Vadim A Shabashov, and Valerii K Kozin. Qdistrnd: A gap package for computing the distance of quantum error-correcting codes. J. Open Source Softw., 7(71):4120, 2022. DOI: 10.21105/joss.04120

    work page doi:10.21105/joss.04120 2022

Showing first 80 references.